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IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003
Improved Gauss–Seidel Projection Method for
Micromagnetics Simulations
Carlos J. García-Cervera and Weinan E
Abstract—The Gauss–Seidel projection method (GSPM)
(Wang et al., J. Comp. Phys., vol. 171, pp. 357–372, 2001) is
a simple, efficient, and unconditionally stable method for micromagnetics simulations. We present an improvement of the
method for small values of the damping parameter. With the new
method, we are able to carry out fully resolved simulations of the
magnetization reversal process in the presence of thermal noise.
Index Terms—Landau–Lifshitz equations, micromagnetics, projection method.
I. INTRODUCTION
U
NDERSTANDING the mechanisms of magnetization
reversal in ferromagnetic samples of nanoscale size is
of interest in the study of the magnetic recording process, in
particular in computer disks and in computer memory cells,
such as MRAMs [1], [2]. Numerical simulation has become an
important tool in the study of both static and dynamic issues in
ferromagnetic materials [3]–[11], and in particular, the magnetization reversal process has been the subject of a large number
of experimental studies. Most studies coincide that the presence
of magnetization vortices inside a ferromagnetic sample has a
dramatic effect in the magnetization reversal process [12]–[16].
It is therefore necessary to resolve numerically length scales
comparable to the size of magnetic vortices in order to carry
out realistic micromagnetics simulations.
The Landau–Lifshitz equation [17], [18]
(1)
describes the relaxation process of the magnetization in a
is the saturation
ferromagnetic material. In (1),
magnetization, and is usually set to be a constant far from
is the permeability of vacuum
the Curie temperature, and
N A in the S.I.). The first term on the
(
being the
right-hand side is the gyromagnetic term, with
gyromagnetic ratio. The second term in the right-hand side is
Manuscript received September 17, 2002; revised January 20, 2003. The work
of W. E was supported in part by the National Science Foundation under Grant
DMS01–30107.
C. J. García-Cervera is with the Mathematics Department, University of California, Santa Barbara, CA 93106 USA (e-mail: cgarcia@math.ucsb.edu).
W. E is with the Mathematics Department and Program in Applied and
Computational Mathematics, Princeton University, Princeton, NJ 08540 USA
(e-mail: weinan@princeton.edu).
Digital Object Identifier 10.1109/TMAG.2003.810610
the damping term, with being the dimensionless damping
coefficient. is the local field
(2)
is the exchange
In (2), is the exchange constant,
is the field
interaction between the spins,
is the external applied field, is
due to material anisotropy,
the volume occupied by the material, and finally, the last term in
(2) is the field induced by the magnetization distribution inside
can be computed
the material. This induced field
by solving
in
outside
(3)
together with the jump conditions
(4)
at the boundary of the domain . In (4), we denote by
jump of at boundary of
the
Currently, the most commonly used methods for the simulation of the Landau–Lifshitz equation are explicit numerical schemes, such as fourth-order Runge–Kutta, or
predictor–corrector schemes, with some kind of adaptive
time stepping procedure. Although explicit schemes may
achieve a high order of accuracy both in space and time, the
time step size is severely constrained by the stability of the
numerical scheme. For physical constants characteristic of the
A m,
J m ,
permalloy (
J m,
T
s ), with a cell
m (256 grid points in a 1- m-long sample),
size
and using fourth-order Runge–Kutta, we need to use a time step
ps for numerical stability. If the
roughly of the order
must
cell size is decreased by a factor of 10, the time step
be reduced by a factor of 100. This time step restriction is even
more severe when considering the stochastic Landau–Lifshitz
equations with Langevin dynamics. In such case, with a cell
m, a time step
fs is needed [19],
size
[20].
A simple, efficient, and unconditionally stable scheme for (1),
the Gauss–Seidel projection method (GSPM), was introduced
0018-9464/03$17.00 © 2003 IEEE
GARCÍA-CERVERA AND E: IMPROVED GAUSS–SEIDEL PROJECTION METHOD FOR MICROMAGNETICS
by Wang et al. in [21]. With this method, we were able to carry
out fully resolved calculations for the switching of the magnetization in micron-size elements. The same method has been used
by Wang et al. to study the dependence of the switching field on
film thickness and the structure of cross-tie walls [22].
The time step size necessary for numerical stability of the
GSPM was found to be independent of the spatial grid size.
was
However, a dependence on the damping parameter
, one can perform
pointed out by Scheinfein [23]. For
ps. For
, one needs
accurate simulations with
to use a time step size ten times smaller. In this paper, we
present an improvement of the GSPM that allows us to use a
ps with
. With the new method,
time step
we have been able to carry out fully resolved simulations of
the stochastic Landau–Lifshitz equations, with time step size
ps, for a cell size
m.
The rest of the paper is organized as follows: In Section II, we
describe the main ideas behind the GSPM. A more detailed description can be found in [21]. The improved GSPM is presented
in Section III. Finally, some numerical results are presented in
Section IV.
II. GAUSS–SEIDEL PROJECTION METHOD
To illustrate the main ideas behind the GSPM, let us consider
first the case when only the exchange term is kept in (2). For
and
clarity, all the constants are ignored. In this case,
the Landau–Lifshitz (1) reduces to
(5)
In order to overcome the stability constraint of explicit schemes,
one usually resorts to implicit schemes [8]. However, due to
the strong nonlinearities present in both the gyromagnetic and
damping terms in the Landau–Lifshitz (1), a direct implicit
discretization of the system is not efficient and is difficult to
implement.
When only the damping term is present, (5) becomes
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easy to implement. It is easy to check that the scheme (8) is
consistent with (7) and is first order accurate in time. However,
direct numerical implementation of (8) shows that the scheme is
unstable. The main idea in the GSPM is to use a Gauss–Seidel
type of technique to improve the stability properties of (8). Let
(9)
The update in the GSPM is defined as follows:
(10)
is used to compute
Note that the newly updated value
, and these two new values are used to update
.
III. IMPROVED GAUSS–SEIDEL PROJECTION SCHEME FOR THE
FULL LANDAU–LIFSHITZ EQUATION
For the full Landau–Lifshitz (1), we couple the above implicit Gauss–Seidel scheme with the projection method developed earlier in [24], in a fractional step framework. Since and
have the same physical dimensions, we can write
,
,
, and
. Without loss
of generality, we will assume that the material is uniaxial, and
, where
,
, and
. Equation (1) is rewritten as
(11)
where
(12)
has dimensions of the reciprocal of time
The constant
). Therefore, we rescale in time
, and we
(
where is the diameter of
rescale the spatial variable
. The equation becomes
(13)
(6)
where
A simple projection scheme for this equation was introduced by
E and Wang [24]. This scheme was shown to be unconditionally
stable and more efficient than other schemes used for the simulation of (6).
The focus in [21] was the gyromagnetic term in the
Landau–Lifshitz equation
(14)
Here, we have defined the dimensionless parameters
, and
.
For our splitting procedure, we define the vector field
(15)
(7)
To overcome the nonlinearity of the equation, consider the
following simple fractional step scheme:
(8)
The advantage of the scheme (8) is that the implicit step is now
linear, comparable to solving heat equations implicitly, and is
We solve
(16)
in three steps.
Step 1) Implicit Gauss–Seidel:
(17)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003
Fig. 1. Simulation of the full Landau–Lifshitz equations using the old GSPM.
Left column: : . Right column: : . For both simulations, x
: m, and t
ps.
0 004
=01
1 =1
= 0 01
1 =
(18)
rotate in any direction. We approximated the stray field by its average value in each cell. This stray field was computed using fast
Fourier transform (FFT). The Helmholtz-type equations that appear in steps 1 and 2 of the GSPM were solved via an FFT-based
fast Poisson solver.
The results of our first experiment are shown in Fig. 1. We
compare the results obtained using the old GSPM for two different values of the damping parameter,
and
,
and without an applied field. For both simulations, we considps and a cell size
m.
ered a time step size
The initial state was fully saturated, and the sample was let
are
to relax without external field. The results for
are
presented in the left column, and the results for
presented in the right column. We show in Fig. 1(a) and (c) a
grayscale plot of the angle between the in-plane magnetization
axis. In Fig. 1(b) and (d), we plot the actual in-plane
and the
magnetization field that corresponds to Fig. 1(a) and (c), respectively. The results obtained with the old GSPM for
suffer a complete loss of accuracy. A time step size
ps
was needed to obtain accurate results. This time step size was
found to be independent of the spatial grid size, confirming the
unconditional stability properties of the numerical scheme.
For our second experiment, we considered the Landau–
Lifshitz equation with stochastic forcing. The Langevin
dynamical equations are
Step 2) Heat flow without constraints:
(19)
(22)
(20)
Step 3) Projection onto
In (22), represents the effect of thermal fluctuations, and it is
uncorrelated, independent Gaussian white noise, characterized
by the moments
:
(21)
(23)
Note that the stray field is recomputed using the intermediate
values of in (17), and (19). In the original implementation of
the GSPM [21], the stray field was not recomputed using these
was used in (17) and
intermediate values. Instead, the value
(19).
IV. NUMERICAL RESULTS
To illustrate the improved properties of the new GSPM,
we have carried out fully resolved simulations of the full
Landau–Lifshitz (1). In our first simulation, we simply illustrate the deficiency of the original GSPM by presenting
.
the results obtained with the original GSPM for
In our second simulation, we have considered the stochastic
Landau–Lifshitz equation with Langevin dynamics and simulated the magnetization reversal process at room temperature.
In our implementation of the GSPM, we divided the computational domain into cells. In each cell, we approximated the
magnetization by a vector with constant magnitude, but free to
represents the expected value of the random variable
In (23),
,
is the Boltzmann constant (
J deg),
is the absolute temperature, and is the gyromagnetic ratio.
represent different spatial locations, and
The subindices in
the superindices represent each vector component of .
The effect of vortices in the magnetization reversal process
has been studied thoroughly in the past few years [16], [14],
[25]. In the experiments presented in [25], the nucleation of
vortices was observed during the reversal process. These vortices facilitated the growth of magnetic domains with increasing
fields, and disappeared after the reversal was complete. The
vortex state was found to be stable, in the sense that the sample
stayed in the vortex state for a wide range of external fields.
In the numerical experiments performed in [25], the magnetization reversal always occurred by vortex nucleation. However,
the vortices only appeared momentarily at the switching field.
For a sample of dimensions 1 m 0.2 m 300 Å, this field
was found to be 190 Oe experimentally and 900 Oe numerically.
GARCÍA-CERVERA AND E: IMPROVED GAUSS–SEIDEL PROJECTION METHOD FOR MICROMAGNETICS
1769
Fig. 2. Comparison of hysteresis loops at zero temperature and at room
temperature, for several values of the damping parameter . The switching
field is approximately 515 Oe at zero temperature. At room temperature, the
reversal occurred at 507 Oe for = 0:001 and = 0:02, and at 503 for
= 0 :1 .
0
0
0
The results at room temperature (
K), for several
values of the damping parameter , are shown in Figs. 2 and 3.
In all cases, we were able to compute the hysteresis loop using
ps, for a cell size
m. For
a time step size
comparison, note that the numerical integration of the stochastic
Landau–Lifshitz equation using the Heun method, for a cell size
m, requires a time step size
fs [19],
[20]. Our method provides an improvement of approximately
four orders of magnitude in the time step size required.
In our simulations, we considered a permalloy sample with
dimensions 1 m 0.2 m 300 Å. A positive external field
was applied to obtain the metastable state shown in Fig. 3(a),
frequently known as S state. The hysteresis loop was calcuOe, the aplated quasi-statically. Starting from a value
plied field was decreased in steps of 4 Oe when equilibrium was
reached. At zero temperature, two criteria were used to determine that a steady state had been reached: either the simulation
had run for 10 ns, or the relative change in the average magnetization after 10 steps was less than 10 . At room temperature,
the simulation was run for 10 ns for each different field.
Our results are consistent with the numerical simulations presented in [25]. The magnetization reversal occurred by vortex
nucleation: a couple of vortices appeared near the end domains,
and they moved quickly, reversing the magnetization. The vortices only appeared momentarily at the switching field. The
switching field obtained in our computations at zero temperOe. At room temperature,
ature was about
Oe. This small
the reversal occurred at about
change in the switching field seems to indicate that thermal effects are not responsible for the nucleation of these vortices.
V. CONCLUSION
We have introduced a simple modification to the GSPM,
which enhances the accuracy and stability properties of the
original method. The new method allows us to perform fully
resolved micromagnetics simulations in the presence of thermal
Fig. 3. Two stages during the magnetization reversal process in a sample
of dimensions 1 m 0.2 m 300 Å, at room temperature. (a) S state,
characterized by a uniform magnetization in the longitudinal direction, with
parallel transverse end domains. (b) Nucleation of vortices near the end
domains. The nucleation field is about 515 Oe.
2
2
0
noise, with realistic values of the damping constant, with a
large time step size.
ACKNOWLEDGMENT
The authors would like to thank M. Scheinfein for pointing
out the problem of the original GSPM. They also thank
A. Arrott, and R. Koch for helpful discussions.
REFERENCES
[1] J. Daughton, “Magnetoresistive memory technology,” Thin Solid Films,
vol. 216, pp. 162–168, 1992.
[2] G. Prinz, “Magnetoelectronics,” Science, vol. 282, p. 1660, 1998.
[3] H. N. Bertram and C. Seberino, “Numerical simulations of hysteresis
in longitudinal magnetic tape,” J. Magn. Magn. Mater., vol. 193, pp.
388–394, 1999.
[4] J. L. Blue and M. R. Scheinfein, “Using multipoles decreases computation time for magnetic self-energy,” IEEE Trans. Magn., vol. 27, pp.
4778–4780, Nov. 1991.
[5] J. Fidler and T. Schrefl, “Micromagnetic modeling—The current state
of the art,” J. Phys. D, Appl. Phys., vol. 33, pp. R135–R156, 2000.
[6] R. Hertel and H. Kronmüller, “Adaptive finite element mesh refinement
techniques in three-dimensional micromagnetic modeling,” IEEE Trans.
Magn., vol. 34, pp. 3922–3930, Nov. 1998.
[7] M. Jones and J. J. Miles, “An accurate and efficient 3D micro-magnetic
simulation of metal evaporated tape,” J. Magn. Magn. Mater., vol. 171,
pp. 190–208, 1997.
1770
[8] Y. Nakatani, Y. Uesake, and N. Hayashi, “Direct solution of the
Landau–Lifshitz–Gilbert equation for micromagnetics,” Jpn. J. Appl.
Phys., vol. 28, no. 12, pp. 2485–2507, 1989.
[9] Y. Nakatani, Y. Uesake, N. Hayashi, and H. Fukushima, “Computer
simulation of thermal fluctuation of fine particle magnetization based
on Langevin equation,” J. Magn. Magn. Mater., vol. 168, pp. 347–351,
1997.
[10] W. Rave, S. Zielke, and A. Hubert, “The magnetic ground state of a
thin-film element,” MPI-MIS Leipzig, Preprint 61/1999.
[11] S. W. Yuan and N. Bertram, “Fast adaptive algorithms for micromagnetics,” IEEE Trans. Magn., vol. 28, pp. 2031–2036, Sept. 1992.
[12] A. F. Popkov, L. L. Savchenko, N. V. Vorotnikova, S. Tehrani, and J. Shi,
“Edge pinning effect in single- and three-layer patterns,” Appl. Phys.
Lett., vol. 77, no. 2, pp. 277–279, 2000.
[13] J. Shi and S. Tehrani, “Edge pinned states in patterned submicron
NiFeCo structures,” Appl. Phys. Lett., vol. 77, no. 11, pp. 1692–1694,
2000.
[14] J. Shi, S. Tehrani, and M. R. Scheinfein, “Geometry dependence of magnetization vortices in patterned submicron NiFe elements,” Appl. Phys.
Lett., vol. 76, pp. 2588–2590, 2000.
[15] J. Shi, T. Zhu, M. Durlam, E. Chen, S. Tehrani, Y. F. Zheng, and J.-G.
Zhu, “End domain states and magnetization reversal in sub-micron magnetic structures,” IEEE Trans. Magn., vol. 34, pp. 997–999, July 1998.
[16] J. Shi, T. Zhu, S. Tehrani, Y. F. Zheng, and J.-G. Zhu, “Magnetization
vortices and anomalous switching in patterned NiFeCo submicron arrays,” Appl. Phys. Lett., vol. 74, pp. 2525–2527, 1999.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003
[17] W. F. Brown Jr, “Interscience tracts on physics and astronomy,” in Micromagnetics. New York: Wiley Interscience, 1963.
[18] L. Landau and E. Lifshitz, “On the theory of the dispersion of magnetic
permeability in ferromagnetic bodies,” Phys. Zeitschrift Sowjetunion,
vol. 8, pp. 153–169, 1935.
[19] P. E. Kloeden and E. Platen, “Applications of mathematics,” in Numerical Solution of Stochastic Differential Equations. Berlin, Germany:
Springer-Verlag, 1992, vol. 23.
[20] T. Schrefl, W. Scholz, D. Süs, and J. Fidler, “Langevin micromagnetics
of recording media using subgrain discretization,” IEEE Trans. Magn.,
vol. 36, pp. 3189–3191, Sept. 2000.
[21] X.-P. Wang, C. J. García Cervera, and W. E, “A Gauss–Seidel projection
method for the Landau–Lifshitz equation,” J. Comp. Phys., vol. 171, pp.
357–372, 2001.
[22] K. Wang, X. P. Wang, and W. E, “Simulation of 3D micro-magnetic
structures,” , to be published.
[23] M. Scheinfein, private communication.
[24] W. E and X. P. Wang, “Numerical methods for the Landau–Lifshitz equation,” SIAM J. Numer. Anal., vol. 38, no. 5, pp. 1647–1665, 2000.
[25] K. J. Kirk, M. R. Scheinfein, J. N. Chapman, S. McVitie, M. F. Gillies,
B. R. Ward, and J. G. Tennant, “Role of vortices in magnetization reversal of rectangular NiFe elements,” J. Phys. D, Appl. Phys., vol. 34,
pp. 160–166, 2001.